Problem: $ C = \left[\begin{array}{rr}-1 & 5 \\ 5 & -2\end{array}\right]$ $ B = \left[\begin{array}{rr}2 & 2 \\ 3 & 1\end{array}\right]$ What is $ C B$ ?
Answer: Because $ C$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C B = \left[\begin{array}{rr}{-1} & {5} \\ {5} & {-2}\end{array}\right] \left[\begin{array}{rr}{2} & \color{#DF0030}{2} \\ {3} & \color{#DF0030}{1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{-1}\cdot{2}+{5}\cdot{3} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{2}+{5}\cdot{3} & ? \\ {5}\cdot{2}+{-2}\cdot{3} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{-1}\cdot{2}+{5}\cdot{3} & {-1}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{1} \\ {5}\cdot{2}+{-2}\cdot{3} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{-1}\cdot{2}+{5}\cdot{3} & {-1}\cdot\color{#DF0030}{2}+{5}\cdot\color{#DF0030}{1} \\ {5}\cdot{2}+{-2}\cdot{3} & {5}\cdot\color{#DF0030}{2}+{-2}\cdot\color{#DF0030}{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}13 & 3 \\ 4 & 8\end{array}\right] $